We present a new approach toward the rational parametrization of canal surfaces. According to our previous work, every canal surface with rational (respectively polynomial) spine curve and rational (respectively polynomial) radius function is a rational (respectively polynomial) Pythagorean hodograph curve in R3,1. Drawing upon this formalism and utilizing the underlying Lorentzian geometry, th...

This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certain r_r matrix are less than 2, where r is the degree of the rational function (or curve), and where the e...

Using homogeneous coordinates, a rational curve can be represented in a nonrational form. Based on such a nonrational representation of a curve, a simple method to identify inflection points and cusps on 2-D and 3-D rational curves is proposed. © 1997 Elsevier Science B.V.

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

Let p ≥ 5 be a prime number and let Fp be a finite field. In this work, we determine the number of rational points on singular curves Ea : y = x(x − a) over Fp for some specific values of a. Keywords—Singular curve, elliptic curve, rational points.

Let be m ∈ Z>0 and a, q ∈ Q. Denote by APm(a, q) the set of rational numbers d such that a, a + q, . . . , a + (m − 1)q form an arithmetic progression in the Edwards curve Ed : x2 +y2 = 1+d x2y2. We study the set APm(a, q) and we parametrize it by the rational points of an algebraic curve.

In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively using of linear fractional transformations between two proper rational parametrizations of the same...

We consider the problem of classifying all univariate polynomials, defined over a domain k, with the property that they and all their derivatives have all their roots in k. This leads to a number of interesting sub-problems such as finding k-rational points on a curve of genus 1 and rational points on a curve of genus 2.

This paper studies the problem of designing a rational Bézier developable surface pencil with a common isogeodesic, and provides an algorithm for the representation of complicated geometric models in industrial applications which need to satisfy that the shape surface can be developed and a given curve is geodesic. By employing the local Frenet orthonormal frame, the explicit expression of the ...

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ǫ > 0 and an ǫ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ǫ-rationality, and we provide an algorithm to parametrize approximately affine ǫ-rational plane curves, without exact singularities at infinit...